Optimal. Leaf size=76 \[ -\frac {x \sqrt {c^2 x^2-1}}{2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {\sqrt {c^2 x^2-1} \tanh ^{-1}(c x)}{2 c d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.20, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {23, 199, 208} \[ \frac {x \sqrt {d-c^2 d x^2}}{2 d^2 \left (1-c^2 x^2\right ) \sqrt {c^2 x^2-1}}+\frac {\sqrt {d-c^2 d x^2} \tanh ^{-1}(c x)}{2 c d^2 \sqrt {c^2 x^2-1}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 199
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \frac {1}{\left (d-c^2 d x^2\right )^2} \, dx}{\sqrt {-1+c^2 x^2}}\\ &=\frac {x \sqrt {d-c^2 d x^2}}{2 d^2 \left (1-c^2 x^2\right ) \sqrt {-1+c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {1}{d-c^2 d x^2} \, dx}{2 d \sqrt {-1+c^2 x^2}}\\ &=\frac {x \sqrt {d-c^2 d x^2}}{2 d^2 \left (1-c^2 x^2\right ) \sqrt {-1+c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \tanh ^{-1}(c x)}{2 c d^2 \sqrt {-1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 57, normalized size = 0.75 \[ \frac {\left (1-c^2 x^2\right ) \tanh ^{-1}(c x)+c x}{2 c d \sqrt {c^2 x^2-1} \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 314, normalized size = 4.13 \[ \left [-\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c x + {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right )}{8 \, {\left (c^{5} d^{2} x^{4} - 2 \, c^{3} d^{2} x^{2} + c d^{2}\right )}}, -\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c x - {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right )}{4 \, {\left (c^{5} d^{2} x^{4} - 2 \, c^{3} d^{2} x^{2} + c d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} \sqrt {c^{2} x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 94, normalized size = 1.24 \[ \frac {\sqrt {-\left (c^{2} x^{2}-1\right ) d}\, \left (-c^{2} x^{2} \ln \left (c x -1\right )+c^{2} x^{2} \ln \left (c x +1\right )-2 c x +\ln \left (c x -1\right )-\ln \left (c x +1\right )\right )}{4 \sqrt {c^{2} x^{2}-1}\, \left (c x +1\right ) \left (c x -1\right ) c \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} \sqrt {c^{2} x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (d-c^2\,d\,x^2\right )}^{3/2}\,\sqrt {c^2\,x^2-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\left (c x - 1\right ) \left (c x + 1\right )} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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